Lecture Notes
POWER ELECTRONICS & MOTION CONTROL I.
Copper electrodesstamped fromcopper ribbon
CathodeAnode
Gate
Planar siliconpellet
Gold connectingwires
Heatsink andanode
Injection mouldedplastic case
2
2
IndexINTRODUCTION _________________________________________________________________________________4
DEFINITION OF POWER ELECTRONICS __________________________________________________________________4INTERDISCIPLINARY CHARACTER OF PE ________________________________________________________________4HISTORICAL BACKGROUND OF DEVICES_________________________________________________________________5CONTRIBUTION OF HUNGARIAN SCIENTISTS IN THE FIELD OF PE ______________________________________________6ECONOMICAL SIGNIFICANCE:_________________________________________________________________________6APPLICATION OF PE________________________________________________________________________________6
Industrial applications ___________________________________________________________________________ 6Residential applications __________________________________________________________________________ 8Commercial Applications _________________________________________________________________________ 8Transportation _________________________________________________________________________________ 8Utility systems__________________________________________________________________________________ 9Aerospace _____________________________________________________________________________________ 9Telecommunications _____________________________________________________________________________ 9Military _______________________________________________________________________________________ 9Computers_____________________________________________________________________________________ 9Medical ______________________________________________________________________________________10
CIRCUITS WITH SWITCHES AND DIODES_________________________________________________________11
SWITCHED DC SOURCE ____________________________________________________________________________11Resistive Load Circuit___________________________________________________________________________11RL Load Circuit _______________________________________________________________________________13Inductive Load Circuit __________________________________________________________________________14
POWER SEMICONDUCTOR SWITCHES ___________________________________________________________16
DIODES ________________________________________________________________________________________16Simplified V-A characteristics ____________________________________________________________________16Schottky diode_________________________________________________________________________________17Fast recovery diode ____________________________________________________________________________17Line frequency diode ___________________________________________________________________________17
THYRISTORS ____________________________________________________________________________________18Thyristor characteristics_________________________________________________________________________18Ways of turning on a thyristor ____________________________________________________________________18Gate triggering of thyristors______________________________________________________________________19
DC Gate Triggering Characteristics _______________________________________________________________________ 20Parameters, Specifications_______________________________________________________________________________ 20Load Lines___________________________________________________________________________________________ 21
Thyristor Turn-Off Characteristics and Methods ______________________________________________________23Turn Off Methods _____________________________________________________________________________________ 23
Application of Thyristors: ON-OFF Control (Burst control) ____________________________________________25Half-wave variable phase control configuration ______________________________________________________________ 25Thyristor in the diagonal of a diode bridge __________________________________________________________________ 26Back-to-back pair of thyristors ___________________________________________________________________________ 27
Chopper Circuits ______________________________________________________________________________28Chopper circuit with auxiliary thyristor ____________________________________________________________________ 28Buck Converter _______________________________________________________________________________________ 30
Parameters, specifications _______________________________________________________________________32Thyristor encapsulations ________________________________________________________________________33
CONTROLLABLE SWITCHES _________________________________________________________________________34BJT (Bipolar Junction Transistor) _________________________________________________________________34MOSFET ( Metal Oxide Semiconductor Field Effect Transistor)__________________________________________35GTO (Gate Turn-Off Thyristor) ___________________________________________________________________35IGBT (Insulated Gate Bipolar Transistor) ___________________________________________________________36Desirable characteristics of a controllable switch _____________________________________________________36Comparison of controllable switches _______________________________________________________________37
POWER DISSIPATION OF A POWER SEMICONDUCTOR SWITCH ________________________________________________37
DC- DC SWITCH MODE CONVERTERS ____________________________________________________________38
3
3
INTRODUCTION __________________________________________________________________________________38CONTROL OF DC-DC CONVERTERS ____________________________________________________________________38STEP-DOWN (BUCK) CONVERTER ____________________________________________________________________40STEP-UP (BOOST) CONVERTER_______________________________________________________________________42CONTINUOUS AND DISCONTINUOUS CONDUCTION MODE (CCM & DCM) _____________________________________43BUCK-BOOST CONVERTER__________________________________________________________________________43CÚK DC-DC CONVERTER___________________________________________________________________________44POSSIBLE WAYS TO CALCULATE THE OUTPUT VOLTAGE RIPPLE:______________________________________________45
4
4
POWER ELECTRONICS & MOTION CONTROL I.
Introduction
Definition of Power ElectronicsTask of Power Electronics (PE) is to control the flow of electric energy by supplying voltages andcurrents in a form that is optimally suited for user loads.
LoadPowerprocessor
Controller
Power input Power output
vi voioii
Reference
MeasurementsControl signals
Block diagram of a power electronic system.
The power processors usually consist of more than one power conversion stage where the operation ofthese stages is decoupled on an instantaneous basis by means of energy storage elements such ascapacitors and inductors.
Power processor
Input OutputConverter 2Converter 1 Energy
storageelement
Power processor block diagram.
Interdisciplinary Character of PEThe study of power electronics encompasses many fields within electrical engineering, as illustrated inthe figure below. These include power systems, solid-state electronics, electrical machines,analog/digital control and signal processing, electromagnetic field calculations, and so on. Combiningthe knowledge of these diverse fields makes the study of power electronics challenging as well asinteresting. There are many potential advances in all these fields that will improve the prospects forapplying power electronics to new applications.
5
5
Powerelectronics
Electromagnetics
Systems andcontroltheory
Electronics
Signalprocessing
Circuittheory
Solid-statephysics
Electricmachines
Powersystems
Simulationand
computing
Interdisciplinary nature of power electronics.
Historical background of devicesThyratron tubesControllable rectifier, basically gas-discharge tube.drawback: limited in current (approx. 50A), limited lifetime ( due to emitting cathode) and highforward voltage drop.
Mercury-Arc Rectifiers
Mercury
C
G
A
Vacuumpump
Current is conducted by the electric arc between the anode (graphite) and the cathode (mercury) whenthe anode is positive as compared to the cathode. Conduction is initiated by ignition through a gatepulse.M-A-Rs made it possible to build large rectifier systems by putting several anodes into a commonmercury tank containing the cathode.Drawback: They need continuous vacuum pumping.
Saturable ReactorsUse the magnetic properties of the core inside a coil.Were replaced by thyristor circuits.Further components:Relays and ContactorsMechanical Speed Changers
6
6
Rheostats e.g. Electric arc welding and motor starters (wet resistance)Constant voltage transformers
Types of Converter systems:1. AC voltage controllers: Fixed voltage AC to variable voltage AC2. Rectifiers: (uncontrolled) Fixed voltage AC to fixed voltage DC3. Rectifiers: (controlled) Fixed voltage AC to variable voltage DC4. DC/DC converters Fixed voltage DC to variable DC (Choppers):
Reduction: Buck ConverterIncrease: Boost ConverterReduce/inc.: Buck-Boost Converter
5. Inverters (uncontrolled.): Fixed DC to fixed AC voltage6. Inverters (controlled): Fixed DC to variable AC voltage (Square/trapezoidal/sine wave output)7. Cycloconverters: Fixed frequency and voltage AC to variable frequency (and voltage)AC
output (frequency reduction)8. Matrix Converters Fixed frequency and voltage AC to variable frequency (and voltage) AC
output (frequency increase/reduction, changing the number of phases)
Contribution of Hungarian Scientists in the field of PE
M. Déry (1854-1938) - 1 phase comutating repulsion motor, AC transformerO.Bláthy (1860-1939) - parallel connection of ac generators, induction current
meter, contribution in development of electric locomotive,AC transformer
Zipernowsky - AC transformerK. Kandó (1869-1931) - development of the first 3 phase high-voltage
locomotive.I. Rácz ( ) - Park vector theory application in PE, particularly in induction motor
drivesF. Csáky (1921-1977) - control theory, automation, power electronics
Economical Significance:Consumption of electric energy in the world is largely in converted form. The next list will showexamples about the application of Power Electronic systems. From the list it is evident that PE haspenetrated almost all fields of technology and our everyday life.
Application of PE
Industrial applications DrivesPumps, Compressors, Blowers and fans, Machine tools, Elevators, Cranes, Conveyors, Hoists.Approx. 2/3 of the generated electric energy in industrial countries is consumed by the various drives.A large portion of these drives is ac drive.
7
7
The basic parts of the drives are the power supplies and the electromechanical power converter(electrical machine.) A functional block diagram of an ac electric drive is seen in the Figure below.
Converter 1 Converter 2Cdc
+
_
acmotor
acac
Utility
Power processor
Figure 3 - Block diagram of an ac motor drive.
Electric heat generation
Arc furnace, Resistive heating , Induction heating (high frequency current in a coil)
HV transmission:
The technical and economical reason for application is that there are certain areas where high voltagetransmission is needed between two systems with different frequencies (e.g.: Japan, where a 50Hzsystem is connected to a 60Hz system) or the HV transmission is more economical as it has higherefficiency (e.g.: HV transmission between Scandinavia and Europe).Used for transmitting high electric power to large distances, through DC line (parallel losses). Thesystem consists of an AC/DC and a DC/AC converter as shown in the figure below.
Power level of such a system is 200-600MW.
Load
High Voltage Line
DC Link
DC/ACconverter
AC/DCconverter
Power SuppliesImportant parameters: Power density [W/in3,W/cm3]
present limits are around 30-50W/in3.
Tolerance Ripple - input reflected ripple (current)
- output ripple (voltage), EMC/EMI - noise levels [dB]
8
8
UPS Systems (Uninterruptable Power Supply) are used where high availability, and/or reliability ofthe power supply is required.
Application: Computers: process control, banks, serversEnergy transmission and distributionCommunicationHospitals
Robots where motors, actuators are needed Induction Heating melting, annealing, heat treatment Electric Arc Welding Power Sources
Residential applications Alarms Refrigeration: temperature control (bang-bang control, on-off control) Space heating Air conditioning Washing machinesdrives
control- program storageelectric heating
Cooking: microwave ovens, induction heating Lighting Electronics (PCs, Home Entertainment) Electric Door Openers Hand Power Tools
Commercial Applications Heating, Ventilating, Air Conditioning (Fan drives) Central Air Conditioning Computers and Office equipment UPS Elevators Lighting electric discharge tubes
CFL (Compact Fluorescent Lamp with integrated PE device)
Transportation Traction Control Battery Chargers (onboard chargers) Electric Locomotives (Super Conducting material, MAGnetic LEVitation vehicle : 500 km/h) Trams, Trolley Buses Subways Automotive electronics including engine control Electric cars (onboard chargers, regenerative braking)
9
9
Utility systems High voltage DC transmission (HVDC) Static var generation (to compensate reactive power in the system to reduce power loss) Nuclear reactor control rod Supplemental energy storage systems (magnetic energy storage with super conducting coil)
stored LW i L Joule
12
2
With super conducting material the current can be raised by magnitudes, which will result inincreased stored energy.
Remote control by audio frequency signals Generator exciters Induced-draft fans and pumps
B
i
F(They deliver the fluid trough the pipe using the interaction of magnetic field and current in the
fluid.)
Active filters, for filtering higher harmonic components time domain control frequency domain control
Aerospace Space shuttle power supply system Satellite power systems, solar power supplies Aircraft power supplies: they are using 400Hz network system. Airports are also using 400Hz
networks.
Telecommunications Battery chargers Power Supplies (DC and UPS) HF inverter radio transmitter for AM service
Military Gun elevation Tracking systems High power density power supplies
Computers Power Supplies (SMPS, e.g. 3.3V, 500A) Floppy and hard disc drives Driving circuits for printer heads
10
10
Medical Artificial internal organs Artificial heart systems HV power supplies for X-ray machines Power supplies for diagnostic equipment (magnetic resonance, computer tomography, ultrasonic
diagnostics, ECG) HF heating (Short wave) Pace makers
11
11
Circuits with Switches and Diodes
In this chapter some circuits made up of ideal elements are discussed. These elements include idealswitches that present either infinite or zero resistance to current and are capable of instantaneoustransition from one state to the other. They also include ideal diodes.
An ideal diode has zero resistance to positive anode current iA, but infinite resistance to current in thereverse direction. Thus the diode conducts if the source voltage v is positive, and the anode-to-cathodevoltage vAC is then zero. The diode does not conduct if the source voltage is negative, when vAC also isnegative. In the diagram of iA versus vAC shown below, the operating point of the diode may thus lie onthe positive axis of iA in the range 0 2 i V RA / or on the negative axis of vAC in the range0 2 v VAK for the circuit of the figure below.
iA
vAC0
2V R
2V
_
R
vAC
+DiA
v V t 2 sin
The purpose of analysing such ideal circuits is to give the reader the ability to look at a circuitembodying power semiconductor devices and to envisage approximately by inspection how that circuitfunctions. This should also show what measures must be taken to protect the practical devices fromdestruction.The operation of a switch in a network may:1. Apply an energy source.2. Remove an energy source.3. Change the configuration of the network in other ways.
In the following sections of this chapter, simple switched circuits are first discussed.
Switched DC sourceThe effect of applying a step function of voltage by means of a switch to circuits of different parametersis discussed in this section. The purpose of doing this is to arrive at conclusions that may be applied tosimilar circuits when they are switched by means of power semiconductor devices.
Resistive Load CircuitIn the circuit below, when switch SW is closed at t=0, the current rises instantaneously to the value
iVR
A
When SW is opened at t=t1, the current falls instantaneously to zero as illustrated. The voltage acrossthe open switch is vS=V.
12
12
V
RV
SW_+
ivR
t1 t0i
V/R
t1 t0
RC Load CircuitIn the circuit below, when SW is closed at t=0, by Kirchhoff’s voltage law
V v vC
idt Ri VC R
t
1
0
1
2.1
Solution of this equation gives an expression for the time variation of current I, and hence also for thevoltages vC and vR. Differentiation of the equation yields
didt RC
i A s 1
0 /
so that i Ae At RC / 2.2
where A is a constant of integration that must be determined from the initial conditions.
V
SW C_ _+ +
i
The capacitor is initially uncharged and therefore has zero potential difference between its plates. Thispotential difference cannot change instantaneously, since
q v Cc
where q is the charge on each plate. For vC to change instantaneously, q must change instantaneously,and this would call for an infinite current. Thus immediately after the switch is closed at t=0+, vC=0,and from equation 2.1
V= vR= Ri [V]so that at t=0+,
I= V/R [A]substitution of t=0 and i= V/R in equation 2.2 yields
A= V/R [A]so that
i VRe A
tRC
2.3
R
t
tvC
i
V
V/R
13
13
and this relationship is shown in the figure. As vR falls, vC rises, until in infinite time (in practicalcircuits often merely a fraction of a second) the capacitor is fully charged, so that
I=0 A: vC= V [V]If the switch were opened at t=t1 before the capacitor was fully charged, then the voltage across theswitch would be
vS= V-vC [V]From equation 2.3 and from the curves of the figure it may be seen that if the resistance in the circuit isvery low, then the initial current may be extremely high, and the flow of current will form a pulse ofvery short duration. If the switch were a power semiconductor, it would be liable to be destroyed by thishigh current.
RL Load Circuitin the circuit below, when SW is closed at t=0
V v v Ldidt
Ri VL R
ordidt
RLi
VL
As
2.4
RV
vLvS
vR
SW L_ _+ +
i
The current in the circuit may be divided into two components. The first is the forced or steady-statecomponent of the current, and this represents the condition of operation of the circuit reached after SWhas been closed for an infinitely long time. It is determined by the applied excitation and is theparticular integral solution of the differential equation describing the circuit. The second component isthe natural or transient component of the current, and this represents a condition of operation of thecircuit that has disappeared after an infinite time. It is determined by the circuit parameters and theinitial conditions existing in the circuit at t=0 and is the complementary function of the solution to thedifferential equation.When the steady-state has been reached, the derivative in equation 2.4 is by definition equal to zero, sothat from equation 2.4 the forced component of the current isiF= V/R [A]The natural component is obtained by solution of the homogeneous equation formed from equation 2.4which is
didt
RLi A sNN 0 /
of which the solution is
t
t
V
0
0
V/R
vL
i
To
14
14
i Ae ANR L t ( / )
where A is a constant of integration that is to be determined. The complete solution of equation 2.4 isthus
i i iVR
Ae AF NR L t ( / ) 2.5
At t=0, i=0, and substitution in equation 2.5 yieldsA= V/R [A]
thus
iVR
e AR L t ( )( / )1
and this function is shown in the figure. The voltage across the inductance is
v Ldidt
Ve VLR L t ( / )
and this function also is shown in the figure.If the switch is reopened, the stored energy in the inductance is released, inducing a voltage at theterminals of the inductance which tends to maintain the current that was flowing while the switch wasclosed. The opening of the switch tends to reduce the current instantaneously to zero, so that di/dtapproaches a value of minus infinity. Since the voltage across the terminals of the inductance is vL=Ldi/dt, this voltage also approaches infinity as indicated in the figure. In the case of a simplemechanical switch operating in atmosphere, the air between the opening contacts is ionized by the fielddue to the high voltage vS and momentarily conducts, forming a high resistance arc in which much ofthe energy formerly stored in the inductor is dissipated as heat. A power transistor operating as a switchwould be destroyed in such a situation.
Inductive Load CircuitIn practical circuits, resistance may be very small and inductance large, so that the result obtained byneglecting resistance in analyzing the circuit approximates closely to its actual behaviour. For such anapproximate circuit shown below, equation 2.4 becomes
didt
VL
A s / 2.6
V
vL
SW L
_+
i
t
t
vLt1
i
(V/L)t1
V
and the time variation of current is that shown in the upper figure, where if SW is closed at t=0, then att=t1
i=(V/L)t1 [A] 2.7
15
15
The problem of how to open the switch without the appearance of an infinite voltage across its contactsremains, and one solution is shown below, where an ideal diode is connected in parallel with theinductance.
VvL
SW
DL _
_
++
i
iL
iD
vD
i t
tiDt1
t1
vL=vD
(V/L)t1
V
t
t
iLt1
t1
(V/L)t1
(V/L)t1
The diode D earns its name “free-wheeling diode” by its ability to permit current iL to continue to flowwhen the energy source has been removed by the opening of the switch at t=t1. The operation of thecircuit for the interval 0<t<t1 is not changed by the addition of the ideal diode and is described byequations 2.6 and 2.7 .For negative values of vL, the inductor is now short circuited by the diode, so that immediately after SWis reopened
v Ldidt
VLL 0
Thus Ldi/dt is zero, and iL=iD continues to flow at the value given by equation 2.7. The voltage vSimmediately after the switch is reopened it is equal to the source voltage V.In practice, the energy stored in the inductance at time t=t1 would be dissipated in whatever resistanceexisted in the inductance-diode mesh of the circuit, and iD would decay exponentially to zero. If theresistance of this mesh were extremely low, then energy would be trapped in the inductance for anappreciable period and might be further increased by a subsequent closing of the switch. This may beclearly seen from a consideration of the current curve in the figure, since if at some instant t2>t1 switchSW were again closed, then the current would once more begin to increase at a constant rate V/L [A/s],starting from the value i=(V/L)t1 [A].
16
16
Power semiconductor switches
Diodes (a switch controlled by the power circuit) Thyristors (basic feature: can be switched on by a signal but only the power circuit can turn it off) Controllable switches (can be switched on and off by a signal)
DiodesStandard symbol:
CAiD
Simplified V-A characteristics
in: 0.1A- few kAisat: A - mAvn: ~1V
iD
isat
in
vr,max vn vD
For the rated current the voltage drop is typically 1V. (by the most types of diodes)The forward characteristics can be approximated by a polynomial.
D D D n Dni a a v a v a v 0 1 2
2 ...
The figure shows the effect of thedifferent polynomial terms to theapproximation of the forwardcharacteristics.
iD
v0 vD
a0+a1vD+a2v D2
a0a0+a1vD
Reverse blocking characteristics
17
17
Region ofavalanche effect
iD
vr,max
vD
saturationcurrent
Reverse recovery charge
iD
t
ideal
real
Qrr
didt
trr
is high
Reverse recovery charge (important, as itcan cause problems by generating over-voltages):
Q i dtrr
trr
D 0
[C]
(It induces high voltage on inductance)
Schottky diodeIt has a low forward voltage drop (0.4V) (Qrr value is very small, practically negligible)Limited blocking voltage ~100V maximum
Fast recovery diodeQrr is small (fast diodes)Due to technological reasons there are basically two types of fast diodes:
in<100A vrated 50500V region Qrr is very lowin200A vrated 5001000V Qrr is moderate
Line frequency diodeRated voltage and current limits are much higher:irated > 1kAvrated > 5kV
18
18
ThyristorsStandard MSZ-EN-symbol (EuroNorm):
AG
C
These devices are capable of blocking voltages in both directions.They have four layers and three junctions.
Cnn pp
A
J3J2J1
G
If vT>0 (forward): J1, J3 are forward biased J2 is reverse biased.
Thyristor characteristics
Ways of turning on a thyristor gating by positive gate current
iL
iH
vF,br vAC
iT
Holdingcurrent
Forwardbreakover
voltage
Gatetriggered
Forward voltage-drop(conducting)
Reverseleakagecurrent
Forwardleakagecurrent
Reversebreakdown
voltage
Latchingcurrent
19
19
by light, in case of Light Activated Thyristors with vF,br (forward breakover voltage), it might destroy the thyristor thermal turn-on (thermal runaway), is normally avoided high dv/dt, fast increase of the voltage vT (it might destroy the thyristor)
The junction of a thyristor can be considered as non-linear capacitance Cj (depends on thevoltage vj).
idqdt
q C v
id C v
dtC
dvdt
vdCdt
Cjj
j j j
Cjj j
jj
jj
( )
tq: turn off time (very important factor of the thyristor)
Unwanted firing through the anode if tqc< tq,device or dvd t
d vd t ra ted
Gate triggering of thyristors
The figure below shows the triggering response of the thyristor device.vD, iD
t
iSt: steadystate currentvS
v0: forwardvoltage drop
90%
10%
iG tR
tD
t
tON: Turn-On timetON=tD+tR
tD: Delay TimeTime interval between the time the gate current pulse reaches 10% of its final value and thetime when the resulting forward current reaches 10% of its maximum value duringswitching from the off-state to the on-state into a resistive load under stated conditions.tR: Rise TimeTime interval between the time the forward current reaches 10% of its maximum value andthe time the forward current reaches 90% of its maximum value during switching from theoff-state to the on-state into a resistive load under stated conditions
20
20
DC Gate Triggering Characteristics
Preferred gatedrive area
Locus of possibletriggering points
Limit lines
Rated peak allowableforward gate voltage
Max allowable instantaneousgate power dissipation
Instantaneousgate voltage
iG
vG
Instantaneous gate current
Parameters, Specifications Instantaneous Forward Gate Current
Instantaneous current flowing between gate and cathode terminals in a direction toforward bias the gate junction.
Instantaneous Forward Gate VoltageInstantaneous forward voltage between gate and cathode terminals with anode terminalopen.
DC Gate Trigger VoltageGate voltage with IGT (DC gate trigger current) flowing but prior to start of anodeconduction.
DC Gate Trigger CurrentForward gate current required to trigger a thyristor at stated temperature conditions.
Peak Reverse Gate VoltageMaximum allowable peak reverse voltage between the gate terminal and the cathodeterminal.
Peak Gate Power DissipationMaximum instantaneous value of gate power dissipation.
Average Gate Power DissipationMaximum allowable value of gate power dissipation averaged over a full cycle.
Holding Current (Gate drive)Value of Instantaneous Forward Current below which thyristor returns to forwardblocking state after having been in forward conduction under stated temperature and gatetermination conditions.
21
21
Latching Current (no Gate drive)Value of minimum anode current to remain in the on-state after removal of the gatetrigger pulse under specified condition.
Instantaneous Reverse Blocking CurrentInstantaneous anode current at stated conditions of negative anode voltage, junctiontemperature, and gate termination.
Instantaneous Forward Blocking CurrentInstantaneous anode current at stated conditions of forward blocking voltage, junctiontemperature, and gate termination.
Load LinesThe trigger circuit load line must intersect the individual thyristor gate characteristic in theregion indicated as “preferred gate drive area”. The intersection, or maximum operatingpoint, should furthermore be located as close to the maximum applicable (peak, average,etc.) gate power dissipation curve as possible. Gate current rise times should be in the orderof several amperes per microsecond in the interest of minimizing anode turn-on timeparticularly when switching into high currents. This in turn results in minimum turn-onanode switching dissipation and minimum jitter.
Trigger circuit shortcircuit current
Trigger circuit opencircuit voltage Maximum
operating point
vOC
iSC
Preferred gatedrive area
SCR characteristics
Load line
Construction of a “load line” is a convenient means of placing the maximum operatingpoint of the trigger circuit-thyristor gate combination into the preferred triggering area. Abasic trigger circuit for driving an thyristor gate is out of a source voltage vS and an internal
22
22
resistance RG. The load line is constructed by connecting a straight line between the triggercircuit open circuit voltage and the short circuit current.If the trigger circuit source voltage is a function of time, the load line sweeps across thegraph, starting as a point at the origin and reaching its maximum position, the load line, atthe peak trigger circuit output voltage.The applicable gate power curve is selected on the basis of whether average or peakallowable gate power dissipation is limiting. For example, if a DC trigger is used, theaverage maximum allowable gate dissipation must not be exceeded. If a trigger pulse isused the peak gate power curve is applicable. For intermediate gate trigger waveforms thelimiting allowable gate power dissipation curve is determined by the duty cycle of thetrigger signal according to:peak gate drive power pulse width pulse repetition rate allowable average gate power
23
23
Thyristor Turn-Off Characteristics and Methods
When a thyristor is in conducting state, there are free moving charge carriers in thejunctions. If we want to block the device (open the switch) we must reduce it’s current tozero. If we have a periodical sine wave current on the device, it will conduct negativecurrent for a time (trr), until the junction region is fully depleted.
_+
iAC
R
vS
vACiG
+
_
iAC
t
vAC
trr
0Qrr
tq
t0
trr: Reverse Recovery TimeThe time interval between zero current and the time at which the reverse current through thedevice has reached a specified value (usually 10% of the peak reverse recovery current)under specified conditions after having been in the on-state.tq: Thyristor (Device) Turn-Off time,a time delay which is defined by the physical device itself. If we apply forward voltage tothe thyristor, before tq is passed, then a current starts to flow (the device goes On-state).It is the time interval between zero current and the time of reapplication of positive forwardblocking voltage under specified conditions with the device remaining in the off-state afterhaving been in the on-state.tqC: Circuit Turn-Off time,the time provided by circuit before positive voltage will be reapplied to the device.tqC tq , must be ensured for safe operation.iRM: Reverse Recovery Current,the current which depletes the junction of charge carriers fully after on-state.
Turn Off Methodsa.)Forced commutation circuit, i=0
R
vd
SW
G
+
_
i
t
v vdtqC
tvF
24
24
Closing the switch will result in iT=0. We must ensure the required tq time.
b.)Reverse voltage supply in the forced commutation circuiti=0, v= -vt
R
vdvt
SW
G
+
+_
_
-vt
i
tv vdtqC
tvF
The opposite direction voltage vt helps clear out the minority carriers of the junctionsection.Below the effect of increasing the voltage vt can be seen.
1/3
tq
1
vt
c.)Line commutation
R
vS G
50Hz tqC1-10ms (ensured by the power circuit)tq=10-100s (device turn-off time)In rectifier operation tqc>tq
vS
t
t
i
G
25
25
Application of Thyristors:ON-OFF Control (Burst control)In this chapter we are going to discuss examples of using thyristors as static switches in linecommutated circuits (turn onelectronic circuit, turn offpower circuit).
With varying of the pulse width (pw) pw=tON/TS the output voltage can be controlled.In this case pulse width (pw) is similar to the duty ratio (D).
D: duty ratio tTON
S
t
OFF ON
TS
tON
V VtTO RMS I RMSON
, ,
PVROUTO RMS
L
,
2
the output power
Half-wave variable phase control configuration
Operation: The gate circuit is driven by the power circuit. A variable resistor is used toadjust the gate current (iG), and so the firing angle (). With increasing the resistor (RG), thegate current will decrease and the firing angle will grow until a maximum angle ofmax=/2. The reason for this is that the falling current gets below the minimal gate current,which is the minimal current required to trigger the thyristor.
RL
RG
iGiL
GT
DA
vi
vG
C
26
26
iv vR
vRL
i F
L
i
L
iV t
RGm
G
sin( )
Turn-on results when iG=iGmin at the firing angle G
i VR
R iVG
m G
GG
G G
m,min
,minsin( ) arcsin
G ,max 2The output voltage (Vo,RMS):
V V t d to RMS m
G
, sin( ) 1
22
and the power dissipation on RL:
PVRouto RMS
L
,2
Full wave operation can be realised by using another (inverse-parallel) thyristor withvariable resistance. The gate resistances are adjusted together.Drawback: High di/dt by firing the thyristor
Causes large RFI (Radio Frequency Interference)
Thyristor in the diagonal of a diode bridge
t
vi, iG
vi
iG
G
iG,mi
iv vR R
as v v and R R ivRG
i G
G Li G G L G
i
G
27
27
D3
T
D4
D2D1
RL
iL
~ vi
Advantage: It provides a full wave rectification with one thyristor.
Back-to-back pair of thyristors
A CG
T2T1SW
AC
RL
iLvi
When a thyristor has a reverse voltage on its terminals (like T2 on the figure), then itscollector-gate terminals can be considered as the terminals of a Zener diode.By closing the switch (SW), the “Zener diode” between C-G of the right (T2) thyristoroperates in the Zener field, and provides the gating signal to T1.
Advantage: average RFI is low in burst-control mode of operation.
28
28
Chopper CircuitsClassification:Configurations, properties:Application :
Chopper circuit with auxiliary thyristor
R1R2
Vd
=
C
T2T1A1
A2
vci1
T1: Main thyristorT2: Commutating (auxiliary) thyristorC: Commutating Condenser
R1<R2, because R1 is the main resistance.
If T1 is conducting and thyristor T2 is fired, then the condenser voltage vC is impressedacross T1 in blocking direction. Thyristor T1 gets in blocking state in a very short time.
Equivalent circuits characterising the operation in a full cycle:
For the analysis of the commutation processes two sections (equivalent circuits) will bestudied.
a.:
+
R1R2
vC
vd
=
C
T2
A2
iC iR2
In case of commutating thyristor T1 (Firing T2):At the end of the commutation iC=0 vC=vd will be the case.
29
29
The current of thyristor T2 is the sum of the load current and the condenser current:
iT2vdR
iC(t) 2
The condenser voltage and current is described by:
vC 2vd 1 e
tT1 vd
iC iT10 e
tT1
These areexponential
functions
b.:
R1R2
vd
=
C
T1
A1 iC
iR1
Commutating thyristor T2 by firing T1 (returning to the on-state):
iTvdR
iC(t)11
30
30
assuming that iC=0 and vC= vd.The condenser voltage and current will be:
vC 2vd 1 e
tT1 vd
iC iT20 e
tT2
The time constants are:T2=R2*C, T1=R1*CR1 is small to produce high load current, thus tq1<tq2.The two resistors must be different, as it is required to adjust the output power. (forexample 3:1 ratio)For reducing losses R2 must be as high as possible, or it has to be part of the load.
Buck ConverterExample of force commutation circuit, where the commutation is practically loss free.(Not used in new designs)
L2D2
C T2 L1
T1iT1
D1 RLVoVd
iL
By changing the duty ratio of T1 we cancontrol the output power.D1: Free wheeling diode (the current ofthe inductor can freely circulate in thecircuit)
TLR1
1
L
D: duty ratio tTON
S
t
OFF ON
TS
tON
Waveforms characterising the operation are shown below:
31
31
t1
Vc
Calculation of the circuit turn-off time, commutating capacitor:
v (t) 1C
i dt vC C C00
tqC
(iC=iL1const, assuming very large inductance)
v (t) 1C
i tC L
When the condenser voltage is zero,t=tqc, Vc=Vd
Vc 1C
i tL qc
The circuit turn-off can be calculated:
tC Viqc
c
L
When T1 is fired, there is a resonant circuit formed by T1-L2-D2-C
L2
vC
+
T L C2 2
With varying the duty ratio of T1 we can set the output voltage VO from 0 up to Vd.Switching frequency is limited by the time t1+T/2.
Disadvantage: second thyristor and a number of components are needed
32
32
due to low frequency a large L1 is required, which is expensive.
Parameters, specifications Repetitive Peak Reverse Voltage
Maximum allowable instantaneous value of repetitive reverse (negative) voltage thatmay be applied to the reverse blocking thyristor anode terminal with gate terminal open.
Non- Repetitive Peak Reverse VoltageMaximum allowable instantaneous value of reverse (negative) voltage including all non-repetitive transient voltages, but excluding all repetitive transient voltages, that may beapplied to the reverse blocking thyristor anode terminal with gate terminal open.
Peak Forward Blocking VoltageMaximum instantaneous value of forward blocking voltage (anode terminal positive)including transient voltages permitted by the manufacturer under stated conditions andwhich will not switch the reverse blocking thyristor to the on-state.
Average Forward Current, On-StateMaximum continuous DC current which may be permitted to flow in the forwarddirection (from anode to cathode) under stated conditions of frequency, temperature,reverse voltage, and current waveform.i v Pave d 0 the dissipated powerv0: Forward Voltage Drop (typically ~1.5V)
Forward Breakover VoltageMaximum positive voltage on the anode terminal with respect to the cathode terminal forwhich the small-signal resistance is zero with stated gate termination. (It is one way toturn on the device. Has to be avoided, most devices will suffer this.)
RMS Forward CurrentMaximum continuous RMS current which may be allowed to flow in the forwarddirection under stated conditions.
Instantaneous Forward Current, On-StateInstantaneous value of anode current flowing into the thyristor in the conducting state.
Instantaneous Forward Voltage DropInstantaneous voltage drop between anode and cathode terminals during conduction ofcurrent from anode to cathode terminals while the device is in the on-state.
Peak One-Cycle Surge Forward CurrentMaximum allowable non-recurrent peak current of a single forward cycle (10milliseconds duration) in a 50 Hz single-phase resistive load system.
I2tThis is a measure of maximum forward non-recurring overcurrent capability for veryshort pulse durations (10 milliseconds or less, unless otherwise specified). I is in RMSamperes, and t is pulse duration in seconds.
Delay Time (tD)Time interval between the time the gate current pulse reaches 10% of its final value andthe time when the resulting forward current reaches 10% of its maximum value duringswitching from the off-state to the on-state into a resistive load under stated conditions.
Rise Time (tR)Time interval between the time the forward current reaches 10% of its maximum value
33
33
and the time the forward current reaches 90% of its maximum value during switchingfrom the off-state to the on-state into a resistive load under stated conditions
Reverse Recovery Time (trr)The time interval between zero current and the time at which the reverse current throughthe device has reached a specified value (usually 10% of the peak reverse recoverycurrent) under specified conditions after having been in the on-state.
Thyristor (Device) Turn-Off time (tq)a time delay which is defined by the physical device itself. If we apply forward voltage tothe thyristor, before tq is passed, then a current starts to flow (the device goes On-state)
Thermal ResistanceTemperature rise per unit power dissipation of a designated point above the temperatureof a stated external reference point under conditions of thermal equilibrium.
RT TPth
J A
tot
Junction TemperatureVirtual junction temperature.Temperature of the pellet, or the hottest layer of the device.(practically the layers havetemperature close to each other)
Case Temperature Ambient Temperature Storage Temperature
Recurrent: When a semiconductor device is applied in such a manner that its maximumallowable peak junction temperature is not exceeded the device is applied on recurrentbasis. Any condition that is normal and repeated part of the application or equipment inwhich the semiconductor device is used must meet this condition if the device is to beapplied on a recurrent basis.Non-recurrent: These ratings allow the maximum (recurrent) operating junction temperatureof the device to be exceeded for a brief instant. This gives the device an instantaneousovercurrent capability allowing it to be coordinated with circuit protective devices.
Thyristor encapsulations Stud type Press pack (hockey pack) Power Block Plastic (TO220)
34
34
Copper electrodesstamped fromcopper ribbon
CathodeAnode
Gate
Planar siliconpellet
Gold connectingwires
Heatsink andanode
Injection mouldedplastic case
Controllable switches
BJT (Bipolar Junction Transistor)is controlled by the base current, has a low current gain.Standard symbol:
B
C
E
Switch frequency range: 25kHz
Characteristics:
By this type of controllable switch the main (collector-emitter current, iCE) current iscontrolled by the base current, iB as shown in the figure beside.Some disadvantages occur in this configuration including slightly higher overall vCE(sat)values and slower switching speeds
35
35
vCE(sat)
i
iC
iB2>iBiB1>iBiB0=0 vCE
normal operatingregion
MOSFET ( Metal Oxide Semiconductor Field Effect Transistor)It is a voltage controlled device. In steady state base current is not required.
Standard symbol:
+
+G
D
SSwitching frequency range: 100kHz3MHzCharacteristics:
iD
vGS=0
vGS=vT
vGS1<0
vGS2< vGS1
vDS
iDS
iDSS
vGS
vT
The drain current, iD is controlled by the gate-source voltage, vGS. The main current reacheszero if the gate-source voltage falls to the voltage vT turn-off voltage.Because of their fast switching speed, the switching losses can be small.Because their on-state resistance has a positive temperature coefficient, MOSFETs areeasily paralleled. This causes the device conducting the higher current to heat up and thusforces it to equitably share its current with the other MOSFETs in parallel.
GTO (Gate Turn-Off Thyristor)Like the thyristor the GTO can be turned on by a short-duration gate current pulse, and oncein the on-state, the GTO may stay on without any further gate current. However, unlike thethyristor, the GTO can be turned off by applying a negative gate-cathode voltage, causing asufficiently large gate current to flow. This negative current need only flow for a fewmicroseconds (during the turn-off time), but it must have a very large magnitude, typicallyas large as one-third the anode current being turned off.
36
36
Device frequency is smaller but its power level is very high Ir=>5kA, Vr=>7kVStandard symbol:
G
A
C
Characteristics:iA
vAK
Turn-On
Turn-Off
The GTO is used when a switch is needed for high voltages and large currents in aswitching frequency range of a few 100 Hz to 10 kHz.Application: traction drives in locomotives
IGBT (Insulated Gate Bipolar Transistor)Similar to the MOSFET, the IGBT has a high impedance gate, which requires only a smallamount of energy to switch the device. Like the BJT, the IGBT has a small on-state voltageeven in devices with large blocking voltage ratings (for example, VON is 2-5 V in a 1000-Vdevice). Similar to the GTO, IGBTs can also be designed to block negative voltages. IGBTshave turn-on and turn-off times on the order of 1 s and are available in module ratings aslarge as 1700 V and 1200 A.Power requirements for the control are very small.Forward voltage drop is much smaller than that of the MOSFET (for high voltageapplications).Symbol:
C
G
E
Characteristics:iC
vGE
0 vCE
Desirable characteristics of a controllable switch
1. Block arbitrarily large forward and reverse voltage.2. Conduct arbitrarily large current with zero voltage drop when on.3. Switch on and off instantaneously when triggered.4. Small power to control.
37
37
Comparison of controllable switches
Device Power Capability Switching SpeedBJT Medium Medium
MOSFET Low FastGTO High SlowIGBT Medium Medium
Power dissipation of a power semiconductor switchThe power dissipation results of several components. By the thyristor: Forward conduction loss (calculated from the forward voltage drop and the forward
current)P v iF F RMS
Switching loss (turn-on, turn-off time)
P W f f V I t V I tSW D SW SW C on C off
12
12max max ( ) max max ( )
Gate loss (on higher frequencies it is important to consider)P v iG G G
Other loss can be for example the reverse loss (calculated from the leakage current and thereverse blocking voltage), but in most cases it is negligible.
38
38
dc- dc switch mode converters
IntroductionThe dc-dc converters are widely used in regulated switch-mode dc power supplies and in dcmotor drive applications. As shown in the figure below, often the input to these convertersis an unregulated dc voltage, which is obtained by rectifying the line voltage, and thereforeit will fluctuate due to changes in the line-voltage magnitude. Switch mode dc-to-dcconverters are used to convert the unregulated dc input into a controlled dc output at adesired voltage level.These converters are very often used with an electrical isolation transformer in the switch-mode dc power supplies and almost always without an isolation transformer in case of dcmotor drives. To discuss these circuits in a generic manner, only the nonisolated convertersare considered in this chapter, since electrical isolation is an added modification.
vcontrol
ACline voltage(1-phase or
3-phase)
DC(unregulated)
DC(unregulated)
DC(regulated)
Battery
FilterCapacitor
UncontrolledDiode
Rectifier
DC-DCConverter Load
In this chapter, the converters are analysed in steady state. The switches are treated as beingideal, and the losses in the inductive and the capacitive elements are neglected. Such lossescan limit the operational capacity of some of these converters and are discussed separately.The dc input voltage to the converters is assumed to have zero internal impedance. It couldbe a battery source; however, in most cases, the input is a diode rectified ac line voltagewitch a large filter capacitance, as shown in the above figure to provide a low internalimpedance and a low-ripple dc voltage source.In the output stage of the converter, a small filter is treated as an integral part of the dc-to-dc converter. The output is assumed to supply a load that can be represented by anequivalent resistance, as is usually the case in switch-mode dc power supplies. A dc motorload ( the other application of these converters) can be represented by a dc voltage in serieswith the motor winding resistance and inductance.
Control of dc-dc convertersIn dc-dc converters, the average dc output voltage must be controlled to equal a desiredlevel, though the input voltage and the output load may fluctuate. Switch-mode dc-dcconverters utilize one or more switches to transform dc from one level to another. In a dc-dcconverter with a given input voltage, the average output voltage is controlled by controllingthe switch on and off durations (tON and tOFF ). To illustrate the switch-mode conversionconcept, consider a basic dc-dc converter shown in the figure below. The average value Voof the output voltage vo depends on ton and toff. One of the methods for controlling theoutput voltage employs switching at a constant frequency (hence, a constant switching time
39
39
period TS=ton+toff) and adjusting the on duration of the switch to control the average outputvoltage. In this method, called pulse-width modulation (PWM) switching, the switch dutyratio D, which is defined as the ratio of the on duration to the switching time period, isvaried.The other control method is more general, where both the switching frequency (and hencethe time period) and the on duration of the switch are varied. Variation in the switchingfrequency makes it difficult to filter the ripple components in the input and the outputwaveforms of the converter.
vo
Vo
+
Vo
Vd
+
R
Vd
toffton
TS
In the PWM switching at a constant switching frequency, the switch control signal, whichcontrols the state (on or off) of the switch, is generated by comparing a signal-level controlvoltage vcontrol with a repetitive waveform as shown below in the block diagram and thecomparator signals of the PWM.
vst = sawtooth voltage
vcontrol(amplified error)
Switchcontrolsignal
vcontrol > vst
vcontrol < vst
(switching frequencyfS= 1/TS)
TS
ton toffOffOff
t0
On On On
vcontrolvcontrol
+Amplifier
Vo (actual)
Vo (desired)
Repetitivewaveform
Switchcontrolsignal
Comparator
Vst
40
40
The control voltage signal generally is obtained by amplifying the error, or the differencebetween the actual output voltage and its desired value. The frequency of the repetitivewaveform with a constant peak, which is shown to be a sawtooth, establishes the switchingfrequency. This frequency is kept constant in a PWM control and is chosen to be in a fewkilohertz to few hundred kilohertz range. When the amplified error signal, which variesvery slowly with time relative to the switching frequency, is greater than the sawtoothwaveform, the switch control signal becomes high, causing the switch to turn on.Otherwise, the switch is off. In terms of vcontrol and the peak of the sawtooth waveform Vstin the figure above, the switch duty ratio can be expressed as
DtonTS
vcontrolVst
Step-down (Buck) converterAs the name implies, a step-down converter produces a lower average output voltage thanthe dc input voltage Vd. Its main application is in regulated dc power supplies and dc motorspeed control.Conceptually, the basic circuit of the figure below constitutes a step-down converter for apurely resistive load. Assuming an ideal switch, a constant instantaneous input voltage Vdand a purely resistive load, the instantaneous output voltage waveform is also shown in thefigure below as a function of the switch position. The average output voltage can becalculated in terms of the switch duty ratio:
VT
v t dtT
V dt dt tT
V D VoS
o
TS
Sd
ton
ton
TSon
Sd d
1 1 00 0
( )
Substituting for D in the above equation yields
VoVdVst
vcontrol k vcontrol
where
kVdVst
constant
41
41
voi
Vo
Vd
0
0
t
io
+
+ C R(load)
L
id
iL
vo=Vovoi vL
+
+
Vd
Frequency spectrumof voi
(=1/TS)
ffS 2fS 3fS
V3fs
V2fs
Vfs
ton toff
TS=1/fS
By varying the duty ratio ton/TS of the switch, Vo can be controlled. Another importantobservation is that the average output voltage Vo varies linearly with the control voltage, asis the case in linear amplifiers. In actual applications, the foregoing circuit has twodrawbacks: (1) In practice the load would be inductive. Even with resistive load, therewould always be certain associated stray inductance. This means that the switch would haveto absorb (or dissipate) the inductive energy and therefore it may be destroyed. (2) Theoutput voltage fluctuates between zero and Vd, which is not acceptable in most applications.The problem of stored inductive energy is overcome by using a diode as shown in the figureabove. The output voltage fluctuations are very much diminished by using a low-pass filter,consisting of an inductor and a capacitor. The above figure also shows the waveform of theinput voi to the low-pass filter , which consists of a dc component Vo, and the harmonics atthe switching frequency fS and its multiples, as shown in the figure. The damping for thelow-pass filter is provided by the load resistor. The corner frequency fc of this low-passfilter is selected to be much lower than the switching frequency, thus essentially eliminatingthe switching frequency ripple in the output voltage.
42
42
During the interval when the switch is on, the diode becomes reverse biased and the inputprovides energy to the load as well as to the inductor. During the interval when the switch isoff, the inductor current flows through the diode, transferring some of its stored energy tothe load.In the steady-state analysis presented here, the filter capacitor at the output is assumed to bevery large, as is normally the case in applications requiring a nearly constant instantaneousoutput voltage vo(t)Vo. The ripple in the capacitor voltage (ouput voltage) is calculatedlater.From the above figure we observe that in a step-down converter, the average inductorcurrent is equal to the average output current Io, since the average capacitor current insteady state is zero.
Step-up (Boost) converter
As the name implies, the output voltage is always greater than the input voltage.
iL
VL
Vd-Vo
Vd
0
t
t
io
+
off
on
D
C R(load)
L
+
Vo
Vd
ton toff
TS=1/fS
When the switch is on, the diode is reverse biased, thus isolating the output stage. The inputsupplies energy to the inductor. When the switch is off, the output stage receives energyfrom the inductor as well as from the input.If D=0 (the duty ratio) then Vo=Vd
43
43
If D is near to 1, then Vo will be very large
Main application of this type is regulated dc power supplies and the regenerative braking ofdc motors.
Continuous and Discontinuous Conduction Mode (CCM & DCM)In these devices there are this two types of operation. In the CCM the output current flowscontinuously, therefore the output voltage is a linear function of the duty ratio. If we are inDCM this means that we are below the boundary current iL,min and the output voltage isn'tanymore a linear function of the duty ratio.
iL
vL
Vd-Vo
0
t
iRL
+
+ C R(load)
L
iL
VovL
+
+
Vd
ton toff
DCM
Boundary betweenthe two modes
CCM
TS=1/fS
There are several applications where it is necessary to avoid the operation in DCM, becausethe output voltage isn't anymore a linear function of the duty ratio and a control circuit isnecessary.
Buck-Boost converter
The main application of a step-down/step-up or buck-boost converter is in regulated dcpower supplies, where a negative-polarity output may be desired with respect to the
44
44
common terminal of the input voltage, and the output voltage can be either higher or lowerthan the input voltage.A buck- boost converter can be obtained by the cascade connection of the two basicconverters: the step-down converter and the step-up converter. In steady-state, the output-to-input voltage conversion ratio is the product of the conversion ratios of the twoconverters in cascade (assuming that switches in both converters have the same duty ratio):
VV
DD
o
d
1
1
This allows the output to be higher or lower than the input voltage, based on the duty ratioD.The cascade connection of the step-down and the step-up converters can be combined intothe single buck-boost converter shown in the figure below.
iL
VL
-Vo
Vd
0
0
t
t
iL
io
+
D
C R(load)
L
+
Vo
Vd
ton toff
TS=1/fS
CCMiL=(iD+io)
CÚK dc-dc converterNamed after its inventor, the Cúk converter is shown in the figure below. This converter isobtained by using the duality principle on the circuit of a buck-boost converter, discussed inthe previous section. Similar to the buck-boost converter, the Cúk converter provides anegative-polarity regulated output voltage with respect to the common terminal of the inputvoltage.
45
45
Its advantage is that because of the two inductors both the input and output currents arecontinuous.
Possible ways to calculate the output voltage ripple:1.) If we need the vo(t) :
- Fourier components are calculated- Y() Attenuated output calculation- summation of the higher harmonic components
2.) If we need only the value of the peak output ripple vo.
ton TS
-Vo
t
t
Vd-Vo
IL2
toff
iL
Q
vo
t
vo(t)
TS/2
t D Toff S ( )1 v tC
i dt QCo C
t
( ) 1
0
Q C Vo
VC
QC
I To
L S 1 1 1
2 2 2
IVL
D TLo
S ( )1 VV
TLC
Do
o
S 2
81( )
The resonance frequency of the filter: fLCC
12
VV
Dff
o
o
C
SW
2 2
21( )
